Solve for $x$ and $y$ using elimination. $\begin{align*}2x-4y &= -3 \\ 6x+9y &= -2\end{align*}$
Solution: We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-3$ and the bottom equation by $1$ $\begin{align*}-6x+12y &= 9\\ 6x+9y &= -2\end{align*}$ Add the top and bottom equations. $21y = 7$ Divide both sides by $21$ and reduce as necessary. $y = \dfrac{1}{3}$ Substitute $\dfrac{1}{3}$ for $y$ in the top equation. $2x-4( \dfrac{1}{3}) = -3$ $2x-\dfrac{4}{3} = -3$ $2x = -\dfrac{5}{3}$ $x = -\dfrac{5}{6}$ The solution is $\enspace x = -\dfrac{5}{6}, \enspace y = \dfrac{1}{3}$.